Testing the Bell inequality with Lambda hyperons April 14, 2009

This morning I came back from Easter vacations to my office and was suddenly assaulted by a pile of errands crying to be evaded, but I prevailed, and I still found some time to get fascinated by browsing through a preprint appeared a week ago on the Arxiv, 0904.1000. The paper, by Xi-Qing Hao, Hong-Wei Ke, Yi-Bing Ding, Peng-Nian Shen, and Xue-Qian Li [wow, I’m past the hard part of this post], is titled “Testing the Bell Inequality at Experiments of High Energy Physics“. Here is the abstract:

Besides using the laser beam, it is very tempting to directly testify the Bell inequality at high energy experiments where the spin correlation is exactly what the original Bell inequality investigates. In this work, we follow the proposal raised in literature and use the successive decays to testify the Bell inequality. […] (We) make a Monte-Carlo simulation of the processes based on the quantum field theory (QFT). Since the underlying theory is QFT, it implies that we pre-admit the validity of quantum picture. Even though the QFT is true, we need to find how big the database should be, so that we can clearly show deviations of the correlation from the Bell inequality determined by the local hidden variable theory. […]

Testing the Bell inequality with the decay of short-lived subatomic particles sounds really cool, doesn’t it ? Or does it ? Unfortunately, my quantum mechanics is too rusty to allow me to get past a careful post which explains things tidily, in the short time left between now and a well-deserved sleep. You can read elsewhere about the Bell inequality, and how it tests whether pure quantum mechanics rules -destroying correlations between quantum systems separated by a space-like interval- or whether a local hidden variable theory holds instead: and besides, almost anybody can write a better account of that than me, so if you feel you can help, you are invited to guest-blog about it here.

Besides embarassing myself, I still wanted to mention the paper today, because the authors make a honest attempt at proposing an experiment which might actually work, and which could avoid some drawbacks of all experimental tests so far attempted, which belong to the realm of quantum optics. In their own words,

Over a half century, many experiments have been carried out […] among them, the polarization entanglement experiments of two-photons and multi-photons attract the widest attention of the physics society. All photon experimental data indicate that the Bell inequality and its extension forms are violated, and the results are fully consistent with the prediction of QM. The consistency can reach as high as 30 standard deviations. […] when analyzing the data, one needs to introduce additional assumptions, so that the requirement of LHVT cannot be completely satisfied. That is why as generally considered, so far, the Bell inequality has not undergone serious test yet.

Being totally ignorant of quantum optics I am willing to buy the above as true, although, being a sceptical son of a bitch, the statement makes me slightly dubious. Anyway, let me get to the point of this post.

Any respectable quantum mechanic could convince you that in order to check the Bell inequality with the decay chain mentioned above, it all boils down to measuring the correlation between the pions emitted in the decay of the Lambda particles, i.e., the polarization of the Lambda baryons: in the end, one just measures one single, clean angle between the observed final state pions. The authors show that this would require about one billion decays of the mesons produced by an electron-positron collider running at 3.09 GeV center-of-mass energy (the mass of the J/psi resonance): this is because the decay chain involving the clean final state is rare: the branching fraction of is 0.013, the decay occurs once in a thousand cases, and finally, each Lambda hyperon has a 64% chance to yield a proton-pion final state. So, 0.013 times 0.001 times 0.64 squared makes a chance about as frequent as a Pope appointment. However, if we had such a sample, here is what we would get:

The plot shows the measured angle between the two charged pions one would obtain from 3382 pion pairs (resulting from a billion decays through double hyperon decay) compared with pure quantum mechanics predictions (the blue line) and by the Bell inequality (the area within the green lines). The simulated events are taken to follow the QM predictions, and such statistics would indeed refute the Bell inequality -although not by a huge statistical margin.

So, the one above is an interesting distribution, but if the paper was all about showing they can compute branching fractions and run a toy Monte Carlo simulation (which even I could do in the time it takes to write a lousy post), it would not be worth much. Instead, they have an improved idea, which is to apply a suitable magnetic field and exploit the anomalous magnetic moment of the Lambda baryons to measure simultaneously their polarization along three independent axes, turning a passive measurement -one involving a check of the decay kinematics of the Lambda particles- into an active one -directly figuring out the polarization. This is a sort of double Stern-Gerlach experiment. Here I would really love to explain what a Stern-Gerlach experiment is, and even more to make sense of the above gibberish, but for today I feel really drained out, and I will just quote the authors again:

One can install two Stern-Gerlach apparatuses at two sides with flexible angles with respect to according to the electron-positron beams. The apparatus provides a non-uniform magnetic field which may decline trajectory of the neutral () due to its non-zero anomalous magnetic moment i.e. the force is proportional to where is the anomalous magnetic moment of , B is a non-uniform external magnetic field and d/n is a directional derivative. Because is neutral, the Lorentz force does not apply, therefore one may expect to use the apparatus to directly measure the polarization […]. But one must first identify the particle flying into the Stern-Gerlach apparatus […]. It can be determined by its decay product […]. Here one only needs the decay product to tag the decaying particle, but does not use it to do kinematic measurements.

I think this idea is brilliant and it might actually be turned into a technical proposal. However, the experimental problems connected to setting up such an apparatus, detecting the golden decays in a huge background of impure quantum states, and capturing Lambdas inside inhomogeneous magnetic fields, are mindboggling: no wonder the authors do not have a Monte Carlo for that. Also, it remains to be seen whether such pains are really called for. If you ask me, quantum mechanics is right, period: why bother ?

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I agree, “why bother testing QM” in this way, but I guess it keeps people employed. As far as Stern-Gerlach experiments, Julian Schwinger invented an elegant formulation of quantum mechanics based on these experiments. He called it the measurement algebra. His first few papers on the subject are fairly readable. There’s also an introductory QM textbook using the method, titled “Quantum Kinematics and Dynamics” by Schwinger.

I would let you know that this idea is not new. I was aware of it some years ago at a conference I attended at Garda lake organized by University of Milan. The last of these conferences was held on 2003 then the organizer, Rodolfo Bonifacio, retired and went to Brazil. The key person behind the idea to use elementary particles to test Bell inequalities is Apollo Go. His homepage at CERN is

Why not not bothering? This is like not bothering for testing if light speed is constant or not at all scales. Indeed, therer are proposals that make QM not work at plank energies, yet, yielding the usual QM rules at lower scales.

I agree that quantum mechanics is right. Still, showing that it is right in very different ways and contexts could bring one some more certainty – or, in the unlikely case, shocking surprise – and maybe some arguments in the neverending struggle against the anti-quantum zealots.